64 DIVISION OF PROPOSITIONS. [CHAP. IV. 



exalted station," and " free from envious regards," is xz. Hence 

 the contrary class, i. e. they to whom this description does not 

 apply, will be represented by 1 - xz, and to this class all men 

 are referred. Hence we have 



y = v(l - xz). 



If the proposition thus expressed had been placed in the equiva- 

 lent form, " Men in exalted stations are not free from envious 

 regards," its expression would have been 



yx - v (1 - z). 



It will hereafter appear that this expression is really equivalent 

 to the previous one, on the particular hypothesis involved, viz., 

 that v is an indefinite class symbol. 



Ex. " No men are heroes but those who unite self-denial to 

 courage." 



Let x = " men," y - " heroes," z = " those who practise self- 

 denial," w 9 "those who possess courage." 



The assertion really is, that " men who do not possess cou- 

 rage and practise self-denial are not heroes." 



Hence we have 



x (1 - zw) = v (1 - y) 



for the equation required. 



15. In closing this Chapter it may be interesting to compare 

 together the great leading types of propositions symbolically ex- 

 pressed. If we agree to represent by X and Y the symbolical 

 expressions of the " terms," or things related, those types will 



be 



X = uY, 



X = Y, 

 vX = vY. 



In the first, the predicate only is particular ; in the second, both 

 terms are universal ; in the third, both are particular. Some mi- 

 nor forms are really included under these. Thus, if Y= 0, the 



second form becomes 



A r =0; 



and if Y= 1 it becomes 



